Salted Apple Pretzel Pie

Grease a very large bowl with butter. Add 1/2 cup warm water (100-115°F) to the bowl of a stand mixer or another large bowl. Sprinkle yeast over. Add brown sugar and stir to dissolve. Let yeast bloom until foamy, 5-7 minutes.

Stir in bread flour, beer, sea salt, and remaining 2 Tbsp. butter with a wooden spoon or spatula. Continue to stir until a shaggy mass forms. Mix with dough hook on medium-low speed until dough forms a smooth ball, about 2 minutes. Dough should be firm and might be a little tacky, but not sticky. If dough is still sticky, add flour, 1 Tbsp. at a time, mixing until smooth. If dough is too dry, add warm water 1 tsp. at a time. Continue to mix on medium-low speed until dough is elastic, 5-7 minutes. (Alternatively, turn out dough onto an unfloured work surface and knead by hand.)

Transfer dough to prepared bowl and cover tightly with plastic wrap. Let dough rise in a warm place until doubled in volume, about 1 1/2 hours.

Meanwhile, preheat oven to 275°F. Line a small rimmed baking sheet or pie pan with foil. Sprinkle baking soda, if using, onto sheet and bake 1 hour. Let cool.

Fill a small stainless steel pot or skillet (not aluminum, copper, or nonstick) with 1 cup water. Add baking soda and bring to a simmer over medium-high heat, stirring to dissolve baking soda. Let cool.

Peel, core, and slice apples into 1/4"-thick wedges. Transfer apples to a large pot and add sugar, all-purpose flour, lemon juice, cinnamon, nutmeg, kosher salt, and 1/4 cup water. Cook over medium heat, stirring occasionally, until apples are softened but still have a little bite, 15-20 minutes. If bottom starts to brown, reduce heat to medium-low and add water, 1 Tbsp. at a time. Let cool to room temperature.

Preheat oven to 450°F. Butter a 10" cast-iron skillet. Punch down dough on an unfloured surface to deflate, then divide in half (an easy way to do this is to use a scale and divide by weight). Return one half to greased bowl and cover with plastic wrap while you work on the other piece.

Using a rolling pin, roll dough to a 13" circle. Settle into prepared skillet and let dough relax into pan, just coming up and over the edge slightly. If it shrinks, stretch it by hand to create an even rim. Cover with plastic wrap and let rise at room temperature until increased in size by about 50%, 20–30 minutes.

Divide second piece of dough into 3 equal pieces. Working with 1 piece at a time and keeping remaining pieces on a rimmed baking sheet covered with plastic, roll on an unfloured surface to form 1 (36"-long) rope. Arrange rope in a U-shape with ends pointing away from you. Holding an end in each hand, cross ends, then fold them down toward you to create a pretzel shape about 11" wide. Return to rimmed baking sheet and cover with plastic wrap. Roll remaining 2 pieces to form 2 more (36"-long) ropes. Twist ropes tightly around each other to make a twisted rope about 35" long. Add to baking sheet with pretzel shape and cover with plastic wrap. Let rise at room temperature until increased in size by about 50%, 20–30 minutes.

If dough has settled into skillet, stretch dough to come back up just to outside rim. Using a fork, evenly prick bottom of dough in pan. Brush dough all over with baking soda mixture, if using. Beat egg yolk with 1 Tbsp. water in a small bowl, then brush dough with egg wash. Cover only outer rim of dough with foil. Bake dough in skillet until light golden brown and dry, 12–15 minutes. Let cool slightly, at least 5 minutes.

Brush just outside rim of crust with egg wash and attach twisted dough rope to create a decorative rim. Fill crust with apple mixture. Center pretzel shape over apple filling, stretching dough until edges of pretzel reach twisted outside rim. Brush twisted dough rope and top of pretzel shape with baking soda mixture, if using, then egg wash. Sprinkle with pretzel salt.

Bake pie until pretzel is golden brown and firm to the touch, 12–15 minutes. If underside of dough seems underbaked, loosely cover top with foil and continue to bake, checking every 2 minutes, until cooked through. Let cool completely before slicing, at least 2 hours.

Pretzel dough can be made 1 day ahead. Let come to room temperature, then let rise in a warm place until doubled in volume, about 1 1/2 hours. Apple pie filling can be made 2 days ahead. Store in a resealable container and chill. Bring to room temperature before baking. Baking soda solution can be made 3 days ahead. Store in a resealable container at room temperature.